Is it permissible for employees to take a loan of 4,000 pounds from a government agency, paid with an administrative fee of 5% upfront (so the employee receives 3,800 pounds), and then repay it over 20 months by deducting 200 pounds monthly, until the full amount (4,000 pounds) is repaid?

The lending entity is not permitted to take any amount from the borrower except for the actual cost of the loan process, without any increase. Otherwise, that would be usury (riba), which is forbidden. Every increase beyond the actual expenses is forbidden. The increase must be only in return for administrative expenses and burdens, and not in return for the loans themselves.